Obvious question: Since the “recess” appts are unconstitutional, shouldn’t the GOP Congressional leadership file the lawsuit?

Republicans agree that Obama’s so-called recess apts yesterday are illegal. So, who has standing to file the lawsuit? I’ve read that the US Chamber of Commerce may do so. But shouldn’t the GOP leadership in Congress..especially the Senate, file the suit to challenge it, and also demand that the Supreme Court take it on an expidited basis?